The dot product has another interesting property with unit vectors.
Imagine that perpendicular to that vector (and through the origin)
passes a plane. Planes divide the entire space into positive
(over the plane) and negative (under the plane), and (contrary to
popular belief) you can also use their math in 2D:

Unit vectors that are perpendicular to a surface (so, they describe the
orientation of the surface) are called unit normal vectors. Though,
usually they are just abbreviated as normals. Normals appear in
planes, 3D geometry (to determine where each face or vertex is siding),
etc. A normalis a unit vector, but it’s called normal
because of its usage. (Just like we call (0,0) the Origin!).

It’s as simple as it looks. The plane passes by the origin and the
surface of it is perpendicular to the unit vector (or normal). The
side towards the vector points to is the positive half-space, while the
other side is the negative half-space. In 3D this is exactly the same,
except that the plane is an infinite surface (imagine an infinite, flat
sheet of paper that you can orient and is pinned to the origin) instead
of a line.

Now that it’s clear what a plane is, let’s go back to the dot product.
The dot product between a unit vector and any point in space
(yes, this time we do dot product between vector and position), returns
the distance from the point to the plane:

GDScript

C#

vardistance=normal.dot(point)

vardistance=normal.Dot(point);

But not just the absolute distance, if the point is in the negative half
space the distance will be negative, too:

I know what you are thinking! So far this is nice, but real planes are
everywhere in space, not only passing through the origin. You want real
plane action and you want it now.

Remember that planes not only split space in two, but they also have
polarity. This means that it is possible to have perfectly overlapping
planes, but their negative and positive half-spaces are swapped.

With this in mind, let’s describe a full plane as a normalN and a
distance from the origin scalar D. Thus, our plane is represented
by N and D. For example:

Basically, N and D can represent any plane in space, be it for 2D or 3D
(depending on the amount of dimensions of N) and the math is the same
for both. It’s the same as before, but D is the distance from the origin
to the plane, travelling in N direction. As an example, imagine you want
to reach a point in the plane, you will just do:

GDScript

C#

varpoint_in_plane=N*D

varpointInPlane=N*D;

This will stretch (resize) the normal vector and make it touch the
plane. This math might seem confusing, but it’s actually much simpler
than it seems. If we want to tell, again, the distance from the point to
the plane, we do the same but adjusting for distance:

GDScript

C#

vardistance=N.dot(point)-D

vardistance=N.Dot(point)-D;

The same thing, using a built-in function:

GDScript

C#

vardistance=plane.distance_to(point)

vardistance=plane.DistanceTo(point);

This will, again, return either a positive or negative distance.

Flipping the polarity of the plane can be done by negating both
N and D. This will result in a plane in the same position, but with
inverted negative and positive half spaces:

So, remember, a plane is just that and its main practical use is
calculating the distance to it. So, why is it useful to calculate the
distance from a point to a plane? It’s extremely useful! Let’s see some
simple examples..

Planes clearly don’t come out of nowhere, so they must be built.
Constructing them in 2D is easy, this can be done from either a normal
(unit vector) and a point, or from two points in space.

In the case of a normal and a point, most of the work is done, as the
normal is already computed, so just calculate D from the dot product of
the normal and the point.

GDScript

C#

varN=normalvarD=normal.dot(point)

varN=normal;varD=normal.Dot(point);

For two points in space, there are actually two planes that pass through
them, sharing the same space but with normal pointing to the opposite
directions. To compute the normal from the two points, the direction
vector must be obtained first, and then it needs to be rotated 90°
degrees to either side:

Here is a simple example of what planes are useful for. Imagine you have
a convex
polygon. For example, a rectangle, a trapezoid, a triangle, or just any
polygon where no faces bend inwards.

For every segment of the polygon, we compute the plane that passes by
that segment. Once we have the list of planes, we can do neat things,
for example checking if a point is inside the polygon.

We go through all planes, if we can find a plane where the distance to
the point is positive, then the point is outside the polygon. If we
can’t, then the point is inside.

Code should be something like this:

GDScript

C#

varinside=trueforpinplanes:# check if distance to plane is positiveif(p.distance_to(point)>0):inside=falsebreak# with one that fails, it's enough

varinside=true;foreach(varpinplanes){// check if distance to plane is positiveif(p.DistanceTo(point)>0){inside=false;break;// with one that fails, it's enough}}

Pretty cool, huh? But this gets much better! With a little more effort,
similar logic will let us know when two convex polygons are overlapping
too. This is called the Separating Axis Theorem (or SAT) and most
physics engines use this to detect collision.

With a point, just checking if a plane
returns a positive distance is enough to tell if the point is outside.
With another polygon, we must find a plane where alltheotherpolygonpoints return a positive distance to it. This check is
performed with the planes of A against the points of B, and then with
the planes of B against the points of A:

Code should be something like this:

GDScript

C#

varoverlapping=trueforpinplanes_of_A:varall_out=trueforvinpoints_of_B:if(p.distance_to(v)<0):all_out=falsebreakif(all_out):# a separating plane was found# do not continue testingoverlapping=falsebreakif(overlapping):# only do this check if no separating plane# was found in planes of Aforpinplanes_of_B:varall_out=trueforvinpoints_of_A:if(p.distance_to(v)<0):all_out=falsebreakif(all_out):overlapping=falsebreakif(overlapping):print("Polygons Collided!")

varoverlapping=true;foreach(PlaneplaneinplanesOfA){varallOut=true;foreach(Vector3pointinpointsOfB){if(plane.DistanceTo(point)<0){allOut=false;break;}}if(allOut){// a separating plane was found// do not continue testingoverlapping=false;break;}}if(overlapping){// only do this check if no separating plane// was found in planes of Aforeach(PlaneplaneinplanesOfB){varallOut=true;foreach(Vector3pointinpointsOfA){if(plane.DistanceTo(point)<0){allOut=false;break;}}if(allOut){overlapping=false;break;}}}if(overlapping){GD.Print("Polygons Collided!");}

As you can see, planes are quite useful, and this is the tip of the
iceberg. You might be wondering what happens with non convex polygons.
This is usually just handled by splitting the concave polygon into
smaller convex polygons, or using a technique such as BSP (which is not
used much nowadays).

This is another bonus bit, a reward for being patient and keeping up
with this long tutorial. Here is another piece of wisdom. This might
not be something with a direct use case (Godot already does collision
detection pretty well) but it’s used by almost all physics engines and collision
detection libraries :)

Remember that converting a convex shape in 2D to an array of 2D planes
was useful for collision detection? You could detect if a point was
inside any convex shape, or if two 2D convex shapes were overlapping.

Well, this works in 3D too, if two 3D polyhedral shapes are colliding,
you won’t be able to find a separating plane. If a separating plane is
found, then the shapes are definitely not colliding.

To refresh a bit a separating plane means that all vertices of polygon A
are in one side of the plane, and all vertices of polygon B are in the
other side. This plane is always one of the face-planes of either
polygon A or polygon B.

In 3D though, there is a problem to this approach, because it is
possible that, in some cases a separating plane can’t be found. This is
an example of such situation:

To avoid it, some extra planes need to be tested as separators, these
planes are the cross product between the edges of polygon A and the
edges of polygon B

So the final algorithm is something like:

GDScript

C#

varoverlapping=trueforpinplanes_of_A:varall_out=trueforvinpoints_of_B:if(p.distance_to(v)<0):all_out=falsebreakif(all_out):# a separating plane was found# do not continue testingoverlapping=falsebreakif(overlapping):# only do this check if no separating plane# was found in planes of Aforpinplanes_of_B:varall_out=trueforvinpoints_of_A:if(p.distance_to(v)<0):all_out=falsebreakif(all_out):overlapping=falsebreakif(overlapping):foreainedges_of_A:forebinedges_of_B:varn=ea.cross(eb)if(n.length()==0):continuevarmax_A=-1e20# tiny numbervarmin_A=1e20# huge number# we are using the dot product directly# so we can map a maximum and minimum range# for each polygon, then check if they# overlap.forvinpoints_of_A:vard=n.dot(v)max_A=max(max_A,d)min_A=min(min_A,d)varmax_B=-1e20# tiny numbervarmin_B=1e20# huge numberforvinpoints_of_B:vard=n.dot(v)max_B=max(max_B,d)min_B=min(min_B,d)if(min_A>max_Bormin_B>max_A):# not overlapping!overlapping=falsebreakif(notoverlapping):breakif(overlapping):print("Polygons collided!")

varoverlapping=true;foreach(PlaneplaneinplanesOfA){varallOut=true;foreach(Vector3pointinpointsOfB){if(plane.DistanceTo(point)<0){allOut=false;break;}}if(allOut){// a separating plane was found// do not continue testingoverlapping=false;break;}}if(overlapping){// only do this check if no separating plane// was found in planes of Aforeach(PlaneplaneinplanesOfB){varallOut=true;foreach(Vector3pointinpointsOfA){if(plane.DistanceTo(point)<0){allOut=false;break;}}if(allOut){overlapping=false;break;}}}if(overlapping){foreach(Vector3edgeAinedgesOfA){foreach(Vector3edgeBinedgesOfB){varnormal=edgeA.Cross(edgeB);if(normal.Length()==0){continue;}varmaxA=float.MinValue;// tiny numbervarminA=float.MaxValue;// huge number// we are using the dot product directly// so we can map a maximum and minimum range// for each polygon, then check if they// overlap.foreach(Vector3pointinpointsOfA){vardistance=normal.Dot(point);maxA=Mathf.Max(maxA,distance);minA=Mathf.Min(minA,distance);}varmaxB=float.MinValue;// tiny numbervarminB=float.MaxValue;// huge numberforeach(Vector3pointinpointsOfB){vardistance=normal.Dot(point);maxB=Mathf.Max(maxB,distance);minB=Mathf.Min(minB,distance);}if(minA>maxB||minB>maxA){// not overlapping!overlapping=false;break;}}if(!overlapping){break;}}}if(overlapping){GD.Print("Polygons Collided!");}